The Impact of Stochastic Matrix on Any Vector

A stochastic matrix shows us that the sum all elements on each column (or each row) is equal one or we can say that each column (or each row) of a stochastic matrix is just a probability vector.

Theorem (1): The sequence of operations a stochastic matrix on any vector will be finally obtained the constant vector. (The elements of vector are the members of Real Number)

V1 = V0 * M^1 , V2 = V1*M^2,  V3 = V2*M^3, …. Vn+1 = V n* M^k

Where vector Vn+1 always will be the constant.

A special state is when the vector is a probability vector which is named Markov Chain or Markov Process

Example (1):

The sample of our vector (1*4) is:

V0 =

                                                                        

Suppose we have below stochastic matrix M (4*4):

M =

The results of the sequences (V0 * M) are as follows:

As we can see, the vectors of V5 and V6 are approximately the same.

Example (2):

The sample of our vector (1*4) is:

V0 = 

The results of the sequences are as follows:

As we can see, the vectors of V5 and V6 are approximately the same.

An example for Markov Chain:

The sample of our probability vector (1*4) is:

V0 = 

I consider above stochastic matrix M (4*4).

Then, we will have below results:

As we can see, the vectors of V5 and V6 are approximately the same.

Is below conjecture is a theorem?

Conjecture (1): The sequence of operations for transpose of a stochastic matrix (n*n) with non - zero elements on any vector will be finally obtained a vector with the same elements. (The elements of vector are the members of Real Number)

Example (1):

Consider above stochastic matrix M (4*4):

M =

The transpose of matrix M is:

^MT =

The sample of vector is:

V0 = 

The results of the sequences are as follows:

As we can see, the elements of vector V7 are the same.

Example (2):

The sample of vector is:

V0 =

The results of the sequences are as follows:

As we can see, the elements of vector V7 are the same.